Writing Exponential Functions

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What is an exponential function?

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An exponential function is a mathematical function of the form f(x) = a(b)^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent.

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What does it mean for a function to be exponential?

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A function is exponential if it grows or decays at a rate proportional to its current value, resulting in a curve that is not linear.

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How do you identify an exponential function from a graph?

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An exponential function graph shows a rapid increase or decrease, forming a J-shaped curve, unlike a straight line for linear functions.

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What is the general form of an exponential growth function?

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The general form is f(t) = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

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What is the general form of an exponential decay function?

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The general form is f(t) = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

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How do you convert a percentage rate to a decimal for exponential functions?

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To convert a percentage to a decimal, divide by 100. For example, 5% becomes 0.05.

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What is the formula for modeling depreciation of an asset?

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The formula is V(t) = V_0(1 - r)^t, where V_0 is the initial value, r is the depreciation rate, and t is time.

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How do you determine the decay rate from an exponential function?

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The decay rate can be found by subtracting the base from 1 in the function f(x) = a(b)^x, where b is the base.

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What is the difference between linear and exponential growth?

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Linear growth increases by a constant amount, while exponential growth increases by a constant percentage of the current value.

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How do you write an exponential function for a situation involving growth?

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Identify the initial amount and the growth rate, then use the formula f(t) = a(1 + r)^t.